For non-negative integers \(m, n\)  and a prime \(p\)  one gets base \(p\)  expansions for \(m\)  and \(n\)  as \(m=\sum \limits _{i=0}^{k}m_ip^i\)  and \(n=\sum \limits _{i=0}^{k}n_ip^i\) . Here we state Lucas theorem ([1]}) which we use extensively in Section 4.
